Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $x = \dfrac{10(5n - 4)}{n} \div \dfrac{50n - 40}{-2} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{10(5n - 4)}{n} \times \dfrac{-2}{50n - 40} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 10(5n - 4) \times -2 } { n \times (50n - 40) } $ $ x = \dfrac {-2 \times 10(5n - 4)} {n \times 10(5n - 4)} $ $ x = \dfrac{-20(5n - 4)}{10n(5n - 4)} $ We can cancel the $5n - 4$ so long as $5n - 4 \neq 0$ Therefore $n \neq \dfrac{4}{5}$ $x = \dfrac{-20 \cancel{(5n - 4})}{10n \cancel{(5n - 4)}} = -\dfrac{20}{10n} = -\dfrac{2}{n} $